Topics in Ultra-cold Bose Gases: the Bose-Hubbard Model; Analogue Models for an Expanding Universe and for an Acoustic Black Hole

In this thesis we consider the application of phase-space methods to Bose-Einstein condensates;\nthe work comprises of three main parts: Part I: A phase-space method for the Bose-Hubbard\nmodel; Part II: An analogue model of an expanding universe in Bose-Einstein condensates. and\nPart III: An analogue model of an acoustic Black Hole in Bose-Einstein condensates.\nIn part I we present a phase-space method for the Bose-Hubbard model based on the Qfunction\nrepresentation. In particular, we consider two model Hamiltonians in the mean-field\napproximation; the first is the standard “one site” model where quantum tunneling is approximated\nentirely using mean-field terms; the second “two site” model explicitly includes tunneling\nbetween two adjacent sites while treating tunneling with other neighbouring sites using the meanfield\napproximation. The ground state is determined by minimising the classical energy functional\nsubject to quantum mechanical constraints, which take the form of uncertainty relations. For each\nmodel Hamiltonian we compare the ground state results from the Q-function method with the\nexact numerical solution. The results from the Q-function method, which are easy to compute,\ngive a good qualitative description of the main features of the Bose-Hubbard model including the\nsuperfluid to Mott insulator. We find the quantum mechanical constraints dominate the problem\nand show there are some limitations of the method particularly in the weak lattice regime.\nAnalogue models of gravity have been motivated by the possibility of investigating phenomena\nnot readily accessible in their cosmological counterparts. In particular, the prediction of quasiparticle\ncreation in ultra-cold Bose gases in specific configurations can be viewed as an analogue to\neither cosmological particle creation or the Hawking effect.\nIn part II of this thesis we investigate the analogue of cosmological particle creation in an\nexpanding universe by numerically simulating a Bose-Einstein condensate with a time-dependent\nscattering length. In particular, we simulate a 2D homogeneous condensate using the classical\nfield method via the truncated Wigner approximation. We show that for several different expansion\nscenarios the calculated particle production is consistent with the underlying theory. For\ninflationary models we find the particle production for long wavelength modes coincides with the\nanalytic theory within the acoustic approximation, whereas the particle production is suppressed\nfor short wavelength (ie. free-particle like) modes. Moreover, particle production is enhanced for\nfaster expansions, approaching the analytic result for the sudden expansion in the limit of a very\nfast expansion. For the case of a cyclic expansion, particle production peaks for a mode frequency\nthat is approximately half of the driving frequency as expected for parametric resonance.\nIn part III of this thesis we investigate an acoustic black hole in a Bose-Einstein condensate,\nformed by two de Laval nozzles in a ring configuration — a system we refer to as the quantum\nde Laval nozzle. Our model is formulated in one dimension with a sinusoidal potential. For nonzero\nsuperfluid flow, this system can exhibit stable transonic flow with both black and white hole\nsonic horizons. Stationary states are found by solving the time-independent Gross-Pitaevskii equation subject to a phase quantisation constraint. By solving the projected Bogoliubov-de Gennes\nequations for the system, we also find the discrete spectrum and quasiparticle modes. There are\ndynamical instabilities for certain values of winding number and potential depth, for which it is\npossible to construct pairs of normalisable modes. We further investigate the dynamics of the\nsystem using a classical field method based on the truncated Wigner approximation. For a low\nwinding number and unstable configuration, we find exponential growth for the pair of unstable\nmodes, whereas there is no growth in these modes for a stable configuration. This can be interpreted\nas non-degenerate parametric amplification, valid for short times. In contrast, for a large\nwinding number, there is significant growth in modes for both stable and unstable configurations.\nThis is indicative of higher order processes neglected in the quasiparticle picture, which is further\nreinforced by that fact that large winding number solutions require large nonlinearities. Finally, we\nconsider the connection of our results with the usual semi-classical prediction of the Hawking effect.\nFor an unstable configuration, the normalised unstable modes couple equal and opposite real\nfrequencies, so that the growth in these modes represents the closest analogy with the Hawking\neffect for our quantum system.

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